It's a temporary permit issued in specific situations where a parent or relative must remain in Italy to care for a minor in the child's best interest, even if other permit conditions wouldn't normally apply.
It can be a lifeline for parents whose normal permits would expire but whose presence is needed for a child's wellbeing. The court usually drives the assessment behind it.
It's usually short-term and granted under specific orders. It's not a substitute for the parent's broader immigration position.
Official source
Ministero dell'Interno
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